What kind of topological invariants can you get from just Hopf algebras?

Question

In A minus sign that used to annoy me but now I know why it is there, Peter Tingley shows how to build knot invariants from the representations of the $U_q(\mathfrak{sl}_2) $ quantum group by comparing it to a certain algebra of ribbons. This is to say the representation theory of this quantum group is "tangled" in some way.

It is known that symmetric functions (characters of the symmetric group $S_n$) form a Hopf algebra. Is there a corresponding topological structure?

I'm just wondering if there's a diagrammatic way to look at representations of $S_n$. In general, does Hopf algebra structure of an algebra, imply there exists a diagrammatic way of looking at its representation theory?

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More examples of Hopf algebras in combinatorics.

Answer 1

The category of finite-dimensional representations of a Hopf algebra has the structure of a monoidal category with duals. The comultiplication induces a monoidal structure, the tensor product of vector spaces. The counit induces a monoidal identity, the ground field. And the antipode induces a duality structure, the vector space dual. Thus, diagrammatically, you can think of intertwiners of representations of a Hopf algebra as being oriented curves with boundary embedded in a strip $\mathbb{R}\times\left[0,1\right]$. The boundary points would be colored by representations, with induced orientations $+$ or $-$ indicating whether we are considering the representation $V$ or its dual $V^\star$. Intertwiners would color the component curves with their orientation determining their source and target endpoints. A quasi-triangular Hopf algebra has an element $R\in H\otimes H$ which induces a braiding on its category of representations. A ribbon Hopf algebra is quasi-triangular and has an element $\theta\in H$ which induces a twist, thus making its category of representations a ribbon category, allowing one to notate intertwiners diagrammatically as colored, framed, oriented tangles.